log-likelihoods

Suppose that the time to failure of an iten of equipment under specified conditions follows an exponential distribution with distribution function , where the expected time to failure is unknown. A life test is carried out in which items are used under conditions as close as possible to those specified. They are installed at different times, so at the end of the test the items have been installed for periods of length . During these intervals, of the items have failed at times (numbering the items according to the time at which failure occurs), while the remaining failure times are censored at .

a) Show that the log-likelihood is given by

,

where denotes the sum of the times for which the items operated before the test ended (either ) or . Show also that the log-likelihood is maximised when equals , i.e. that the maximum-likelihood estimate of is ; find an expression for the maximum value.

b) The results of a life test on 10 items of coal mining equipment are given below, all times being measured from the day of installation of the first item. Assuming an exponential distribution of time to failure, calculate the log-likelihoods and the maximum-likelihood estimate of .